Weak solutions of semilinear elliptic equation involving Dirac mass

Abstract

In this paper, we study the elliptic problem with Dirac mass(1){−Δu=Vup+kδ0inRN,lim|x|→+∞⁡u(x)=0, where N>2, p>0, k>0, δ0 is the Dirac mass at the origin and the potential V is locally Lipchitz continuous in RN∖{0}, with non-empty support and satisfying0≤V(x)≤σ1|x|a0(1+|x|a∞−a0), with a0<N, a0<a∞ and σ1>0. We obtain two positive solutions of (1) with additional conditions for parameters on a∞,a0, p and k. The first solution is a minimal positive solution and the second solution is constructed via Mountain Pass Theorem.